# Math Monday: Try a Torus

By Glen Whitney for the Museum of Mathematics

Just suppose you wanted to make your own model of the Ungar-Leech map on the surface of a torus, like this one created by Norton Starr in 1972:

You’d probably want to start by making a torus. Suppose you didn’t happen to have access to a FabLab or a Modela MDX milling machine, so that you couldn’t follow these instructions to produce a wooden torus like this one:

What could you do? Well, you could try making a plaster mold of a torus, as in the following detailed video showing the entire process from start to finish:

And if you do make a torus in this way, you really might want to paint it with the Ungar-Leech map shown above. Why? Because that map shows that unlike on a sphere, where any map can be colored with four colors, it takes at least seven colors to color certain maps on a torus. In particular, the Ungar-Leech map divides the torus into seven congruent regions, each of which touches all of the other six. So there’s no way to color it with fewer than seven colors.

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### Gareth Branwyn

Gareth Branwyn is a freelance writer and the former Editorial Director of Maker Media. He is the author or editor of over a dozen books on technology, DIY, and geek culture. He is currently a contributor to Boing Boing, Wink Books, and Wink Fun. His free weekly-ish maker tips newsletter can be found at garstipsandtools.com.

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